Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets

Summary.

The cascade algorithm with mask a and dilation M generates a sequence $\phi_n, n=1,2,\ldots,$ by the iterative process

\[ \phi_n(x) = \sum_{\alpha\in{\mathbb Z}^s} a(\alpha) \phi_{n-1}(Mx - \alpha) \quad x\in{\mathbb R}^s, \]

from a starting function $\phi_0,$ where M is a dilation matrix. A complete characterization is given for the strong convergence of cascade algorithms in Sobolev spaces for the case in which M is isotropic. The results on the convergence of cascade algorithms are used to deduce simple conditions for the computation of integrals of products of derivatives of refinable functions and wavelets.